Settings. For an \(N_C\)  class problem, we assume that our samples are drawn from a product space \(\mathcal {Z} = \mathcal {X} \times \mathcal {Y}\) . \(\mathcal {X}\)  is the input feature where \(\mathcal {X} \subset \mathbb {R}^d\)  and \(d\)  is the input dimensionality. \(\mathcal {Y}\)  is the label space \([N_C]\) . Given a label \(y_m= i\) , we will use the one-hot vector \(\mathbf {y}_m= [y^{(1)}_m,\cdots ,y^{(N_C)}_{m}]\)  to represent it, with \(y^{(k)}_{m} = 0, k \ne i\) , and \(y^{(k)}_m = 1\)  if \(k = i\) . In this paper, we will adopt the one vs. all decomposition [1]} of the multiclass problem. Under this context, an \(N_C\) -class (\(N_C>2\) ) scoring function refers to a set of \(N_C\)  functions \(f = (f^{(1)}, \cdots , f^{(N_C)})\) , where \(f^{(i)}: \mathcal {X} \rightarrow \mathbb {R}\)  serves as a continuous score function supporting \(y=i\) .
